cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A357265 Expansion of e.g.f. -LambertW(x * log(1-x)).

Original entry on oeis.org

0, 0, 2, 3, 32, 150, 1884, 16380, 249808, 3255336, 59596560, 1037413080, 22432698144, 486784686960, 12233449250736, 316660035739320, 9111729094222080, 273147758526888000, 8880267446524694016, 301952732236006556160, 10963551960785051470080
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2022

Keywords

Links

Crossrefs

Cf. A052807, A355842, A357267.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-lambertw(x*log(1-x)))))
  • PARI
    a(n) = n!*sum(k=1, n\2, k^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=1..floor(n/2)} k^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

A356752 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2/2 * A(x)).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 360, 2394, 17220, 260280, 3076920, 35980560, 595686960, 9760411440, 159321570408, 3093987619800, 63314740616400, 1318245318411840, 30240056863978560, 736919729169603840, 18522487833889334400, 495842871278901363840, 14014346231616983128800
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A351492, A356912.
Cf. A355842, A356753.

Programs

  • Mathematica
    nmax = 22; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^2/2*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^2/2*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2/2*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2/2*log(1-x))/(x^2/2*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^2/2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^2/2 * log(1-x)) ).
E.g.f.: A(x) = LambertW(x^2/2 * log(1-x))/(x^2/2 * log(1-x)).

A356753 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3/6 * A(x)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 3024, 25200, 225000, 2217600, 29974560, 400720320, 5558957040, 81340459200, 1344965825280, 23566775232000, 432681781459200, 8309927446329600, 170258024427580800, 3679448236206220800, 83235946152090547200, 1962840630226968307200
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A351493, A356913.
Cf. A355842, A356752.

Programs

  • Mathematica
    nmax = 23; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^3/6*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^3/6*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*log(1-x))/(x^3/6*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^3/6 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3/6 * log(1-x)) ).
E.g.f.: A(x) = LambertW(x^3/6 * log(1-x))/(x^3/6 * log(1-x)).

A356905 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.

Original entry on oeis.org

1, 0, 2, 3, -4, -30, 294, 3780, -7904, -444528, 78840, 99657360, 539299848, -27852945120, -361237078944, 10124338180320, 258341121976320, -4020500134465920, -205187357182405824, 1330097523844832640, 186823640933648588160, 500469438126120583680
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A355842, A356795, A356796, A356906.
Cf. A349561, A356884.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = (1/(1 - x)^x)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*log(1-x)/lambertw(-x*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x * log(1-x)) ).
E.g.f.: A(x) = -x * log(1-x)/LambertW(-x * log(1-x)).

A356795 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 68, 330, 7674, 73080, 1883440, 28281960, 818625960, 17120406600, 557507325000, 15014517495120, 548643259812816, 18056683281775320, 736892260092195840, 28579282973977498560, 1295028345251832359616, 57666859088090317591680
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A066166, A355842, A356796.
Cf. A356786.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*log(1-x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*log(1-x))/(2*x*log(1-x)))^(1/2)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(2 * x * log(1-x))/2 ).
E.g.f.: A(x) = ( LambertW(2 * x * log(1-x))/(2 * x * log(1-x)) )^(1/2).

A356796 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^3).

Original entry on oeis.org

1, 0, 2, 3, 92, 450, 14454, 141540, 4980128, 78711696, 3048567480, 68677353360, 2930551701384, 86832573553440, 4079649847428960, 150444517302424800, 7768028697749806080, 342721736137376184960, 19392702029822685015360, 994397473912386435004800
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A066166, A355842, A356795.
Cf. A356787.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*log(1-x))/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*log(1-x))/(3*x*log(1-x)))^(1/3)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (3*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(3 * x * log(1-x))/3 ).
E.g.f.: A(x) = ( LambertW(3 * x * log(1-x))/(3 * x * log(1-x)) )^(1/3).

A356906 E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x)^x.

Original entry on oeis.org

1, 0, 2, 3, -28, -150, 2154, 26040, -322512, -7872984, 77570280, 3752301960, -22068935736, -2542757920560, 1422846762960, 2302464947491800, 14860063644794880, -2653728770258072640, -41790782141846648640, 3739260018343338345600
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Crossrefs

Cf. A355842, A356795, A356796, A356905.
Cf. A349652, A356885.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (-2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-2*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*x*log(1-x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*log(1-x)/lambertw(-2*x*log(1-x)))^(1/2)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-2*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-2 * x * log(1-x))/2 ).
E.g.f.: A(x) = ( -2 * x * log(1-x)/LambertW(-2 * x * log(1-x)) )^(1/2).
Showing 1-7 of 7 results.

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